duality in physics, not tri-ality, implies that only Riemannian and Lobachevskian geometries exist and that Euclidean is not a geometry

From: Archimedes Plutonium (a_plutonium_at_iw.net)
Date: 03/21/05 

Date: Mon, 21 Mar 2005 14:15:11 -0600

I am in the midst of cleaning up my website of www.iw.net/~a_plutonium
after approx 13 years of neglect. I have come a long way since my first
posts of August 1993 to the sci newsgroups.

In the 1990s I had the instinct and suspicion that the Doubly Infinites
were Lobachevskian geometry and that the Adics formed the intrinsic
points of Riemannian geometry and that the REals formed the intrinsic
points of Euclidean Geometry.

But this is 2005 and after I unified the forces of physics a few years
back I clung to the idea that Duality of Physics tells me there are only
2 real geometries and that the Euclidean Geometry was a illusion or a
set that was not formed. That Euclidean Geometry involves only a point,
a single point of zero and it does not exist in reality. Physics is
duality and that means there exists only 2 geometries, not 3.

And that means there exists only 2 Number Systems. One is infinite
strings rightward which we know of as Reals and infinite strings
leftward which we know of as Adics. And the Adics form Riemannian
geometry.

But what geometry do the REals form? They do not form Euclidean because
that is a defunct set which contains only one element of zero. That
leaves the REals as forming the geometry of Lobachevsky with its
negative curvature.

However, not all is lost with Euclidean geometry. It maybe the case that
those Doubly Infinites that were discussed in 1993 with Karl Heuer and
myself, maybe just the points of Euclidean Geometry.

So below is what the file 109 presently contains but which I will edit
and remove some old posts:

File 109 Analytical Riemannian Geometry; Analytical Lobachevskian
Geometry

Subject: ANALYTICAL RIEMANNIAN GEOMETRY & ANALYTICAL
LOBACHEVSKIAN GEOMETRY
From: Archimedes.Plutonium@dartmouth.edu (Archimedes Plutonium)
Date: 1997/05/10
Message-Id: (5l2ff3$1ui$1@dartvax.dartmouth.edu>
Newsgroups: sci.physics,sci.math,sci.logic,sci.physics.electromag

In article (5l0mq8$1e9$1@dartvax.dartmouth.edu>
Archimedes.Plutonium@dartmouth.edu (Archimedes Plutonium) writes:

> All of the above is an aside to the project at hand to discover the
> intrinsic coordinate systems of Riemannian and Lobachevskian
geometries
> and to discover ANALYTICAL RIEMANNIAN GEOMETRY and ANALYTICAL
> LOBACHEVSKIAN GEOMETRY.

  The above is one and the same, if I discover the intrinsic coordinate
system of Riemannian geometry then I know Analytical Riemannian
geometry, or, vice versa.

[lines deleted]

 So you can sort of appreciate
what I am encountering when I tell people that there exists two other
coordinate systems besides the Cartesian coordinate system.

  But let me summarize here and now what has come to pass on this
project. In 1637 mathematics can assert that the world had Analytical
Euclidean Geometry with Descartes. And this is commonly called the
Cartesian coordinate system. It is the Reals and for 2nd dimension an
appended imaginary number i which is really a 90 degree angle. And for
3rd dimension another appended imaginary number called j. A few
comments before I leave the year 1637. It was really called Analytical
Geometry and not Analytical Euclidean Geometry. For in 1637 until circa
1823 with Bolyai and Lobachevsky did the world come to start to
recognize that there existed three geometries. Unfortunately to the
present day Analytical Geometry is still thought of as just one subject
and that the Cartesian coordinate system suffices for Riemannian and
Lobachevskian Geometries.
   Part of the blame lies in Number theory that no-one has opened the
idea that Riemannian and Lobachevskian geometries have their own
special numbers and their own special coordinate system. The big
problem here in mathematics is a teaching problem. Throughout the
world, ever since Gauss it has been taught to students of mathematics
that all of math is begot from the Natural Numbers = Finite Integers.
When you teach all students of mathematics that the foundation of
numbers is Finite Integers, then of course you will accept without
question that the Cartesian coordinate system is the only coordinate
system in existence and that Riemannian and Lobachevskian geometry do
not have their own coordinate system so why even think that they do.
>From Gauss to the present time, all of mathematics was built from
Naturals = Finite Integers, even geometry. And as one notable
mathematician remarked words to the effect, " God made the Naturals,
all the rest of mathematics is man made."
   It is alien to most every present day mathematician to consider my
theses that just as there exists 3 independent geometries, there exists
3 independent Number systems and hence 3 independent Number theories.
And worst of all, mathematics does not start with Naturals = Finite
Integers, for that set is not even a mathematical entity.

  Take all of the Reals and lop off the rightward infinite string, and
what is remaining (not considering the negative Reals), are the
Naturals, the Finite Integers.

   Take all of the P-adic Numbers and lop off the infinite leftward
string. What is remaining is the finite portion. Are we to say that
this finite portion holds great significance to mathematics? Well,
ludicrously since Gauss we have said that all of mathematics rests on
the finite string of the Reals.

  Then by 1900 Kurt Hensel discovered the P-adics. And of course, since
the Naturals = Finite Integers were in absolute tryannical rule , it
was thought that these p-adics were just some sort of oddity, some
extension of the Naturals.

   Then by 1991, Archimedes Plutonium via intuition said that the
Riemann and Lobachevsky geometries have their own special coordinate
system, distinct and independent of the Cartesian coordinate system for
Analytical Euclidean Geometry.

   By 1993, I had independently discovered the Infinite Integers and
later in that same year I was to learn that the p-adics were these
Infinite Integers. In the end of 1993, I conjectured that the P-adics
were the intrinsic numbers of Riemannian Geometry and that the
heretofore unknown species of numbers which are infinite strings both
rightwards and leftwards, which I called Doubly Infinites, were the
intrinsic numbers of Loba geometry. I was not sure in 1993 or the years
that passed until 1997, whether the Riem geometry was the P-adics or
the Doubly Infinites.

   By May of 1997, I settled on the idea that the Doubly Infinites were
Riemannian Geometry. I had made little progress between 1993 up until
April 1997 because I did not have the proper model to help me. In May
of 1997 I was hoping to have a model. It is a sinusoid curve in the
first quadrant
that is revitalizing and boosting the progress of discovering
Analytical Riem/Loba Geometries.

[lines deleted]

  So here is the summary as of this writing.

 I get Loba geometry in 2nd dimension with the half-circles in the
half-plane.

Any two points in the half-plane determine a unique semi-circle, easy
rule from High School geometry. Given two arbitrary points in the
half-plane connect those two points by a line segment. Take the
midpoint of that line segment and draw the perpendicular, and where it
intersects the x-axis is the center of a circle. Thus draw the
semi-circle.

 Can I get Riem geometry in 2nd dimension by Tangent Sinusoid??

  x sin(A x) the Tangent Sinusoid ??

----------------------------------------------------

Subject: Re: ANALYTICAL RIEMANNIAN GEOMETRY & ANALYTICAL
LOBACHEVSKIAN GEOMETRY
From: Archimedes.Plutonium@dartmouth.edu (Archimedes Plutonium)
Date: 1997/05/14
Message-Id: (5ld52s$c6n$1@dartvax.dartmouth.edu>
Newsgroups: sci.physics,sci.math,sci.logic,sci.physics.electromag

Are springs and DNA helix the

Natural Architecture of the Adics

 [from a conversation with a good friend of mine]

>Do the adics have a bend in them that comes back around to 0?

Not curvature, no. The "shape" of the adic numbers is a tree rather
than
a line.

Reals: 1 --- 2 --- 3 --- 4 --- 5 --- 6 --- 7

2-adics:
      4
    / \
  2 6
 / \ / \
1 3 5 7

Somebody once said that the tree of evolution has a natural tendency
to be p-adic. (I probably read that on talk.origins.)

Actually, maybe this structure is more meaningful. This is a tree of
successive approximations in the 2-adics.

         /---0-----\
        / / \ \
       / / 1--\ \
      / / / \ \ \
     / 2 / 3 \ \
    / /| / / \ \ \
   4 6 | 5 / 7 \ \
  / / | | | | \ \
12 14 10 13 11 15 9 8
-------------------------------------------------------------

Subject: Re: Natural Architecture of adics

 [from a conversation with a good friend of mine]

In my earlier diagram, I had the wrong values on the nodes of the tree.
The one I gave later in the message was better, but confusing because
of
the branches that crossed more than one level. I should have
duplicated
the nodes at the lower level, instead:

                 0
             / \
         0 1
       / \ / \
     0 2 1 3
    / \ / \ / \ / \
   0 4 2 6 1 5 3 7

Now, due to limited screen width, it will be more convenient to draw
the
trees sideways, with the root node on the left and the different levels
in columns to the right. In the n-adic numbers, this will be an n-ary
tree; each node has n children which are written in the column
immediately
to the right and down. Here's that same tree in this notation:

   0 0 0 0
               4
           2 2
               6
       1 1 1
               5
           3 3
               7

Now we have enough room to show them to cardinality around 100.
(Actually I'll use the nearest power of n.)

2-adic tree with cardinality 128:
   0 0 0 0 0 0 0 0
                              64
                          32 32
                              96
                      16 16 16
                              80
                          48 48
                             112
                   8 8 8 8
                              72
                          40 40
                             104
                      24 24 24
                              88
                          56 56
                             120
               4 4 4 4 4
                              68
                          36 36
                             100
                      20 20 20
                              84
                          52 52
                             116
                  12 12 12 12
                              76
                          44 44
                             108
                      28 28 28
                              92
                          60 60
                             124
           2 2 2 2 2 2
                              66
                          34 34
                              98
                      18 18 18
                              82
                          50 50
                             114
                  10 10 10 10
                              74
                          42 42
                             106
                      26 26 26
                              90
                          58 58
                             122
               6 6 6 6 6
                              70
                          38 38
                             102
                      22 22 22
                              86
                          54 54
                             118
                  14 14 14 14
                              78
                          46 46
                             110
                      30 30 30
                              94
                          62 62
                             126
       1 1 1 1 1 1 1
                              65
                          33 33
                              97
                      17 17 17
                              81
                          49 49
                             113
                   9 9 9 9
                              73
                          41 41
                             105
                      25 25 25
                              89
                          57 57
                             121
               5 5 5 5 5
                              69
                          37 37
                             101
                      21 21 21
                              85
                          53 53
                             117
                  13 13 13 13
                              77
                          45 45
                             109
                      29 29 29
                              93
                          61 61
                             125
           3 3 3 3 3 3
                              67
                          35 35
                              99
                      19 19 19
                              83
                          51 51
                             115
                  11 11 11 11
                              75
                          43 43
                             107
                      27 27 27
                              91
                          59 59
                             123
               7 7 7 7 7
                              71
                          39 39
                             103
                      23 23 23
                              87
                          55 55
                             119
                  15 15 15 15
                              79
                          47 47
                             111
                      31 31 31
                              95
                          63 63
                             127

3-adic tree with cardinality 81:
   0 0 0 0 0
                  27
                  54
               9 9
                  36
                  63
              18 18
                  45
                  72
           3 3 3
                  30
                  57
              12 12
                  39
                  66
              21 21
                  48
                  75
           6 6 6
                  33
                  60
              15 15
                  42
                  69
              24 24
                  51
                  78
       1 1 1 1
                  28
                  55
              10 10
                  37
                  64
              19 19
                  46
                  73
           4 4 4
                  31
                  58
              13 13
                  40
                  67
              22 22
                  49
                  76
           7 7 7
                  34
                  61
              16 16
                  43
                  70
              25 25
                  52
                  79
       2 2 2 2
                  29
                  56
              11 11
                  38
                  65
              20 20
                  47
                  74
           5 5 5
                  32
                  59
              14 14
                  41
                  68
              23 23
                  50
                  77
           8 8 8
                  35
                  62
              17 17
                  44
                  71
              26 26
                  53
                  80

5-adic tree with cardinality 125:
   0 0 0 0
              25
              50
              75
             100
           5 5
              30
              55
              80
             105
          10 10
              35
              60
              85
             110
          15 15
              40
              65
              90
             115
          20 20
              45
              70
              95
             120
       1 1 1
              26
              51
              76
             101
           6 6
              31
              56
              81
             106
          11 11
              36
              61
              86
             111
          16 16
              41
              66
              91
             116
          21 21
              46
              71
              96
             121
       2 2 2
              27
              52
              77
             102
           7 7
              32
              57
              82
             107
          12 12
              37
              62
              87
             112
          17 17
              42
              67
              92
             117
          22 22
              47
              72
              97
             122
       3 3 3
              28
              53
              78
             103
           8 8
              33
              58
              83
             108
          13 13
              38
              63
              88
             113
          18 18
              43
              68
              93
             118
          23 23
              48
              73
              98
             123
       4 4 4
              29
              54
              79
             104
           9 9
              34
              59
              84
             109
          14 14
              39
              64
              89
             114
          19 19
              44
              69
              94
             119
          24 24
              49
              74
              99
             124

10-adic tree with cardinality 100:
   0 0 0
          10
          20
          30
          40
          50
          60
          70
          80
          90
       1 1
          11
          21
          31
          41
          51
          61
          71
          81
          91
       2 2
          12
          22
          32
          42
          52
          62
          72
          82
          92
       3 3
          13
          23
          33
          43
          53
          63
          73
          83
          93
       4 4
          14
          24
          34
          44
          54
          64
          74
          84
          94
       5 5
          15
          25
          35
          45
          55
          65
          75
          85
          95
       6 6
          16
          26
          36
          46
          56
          66
          76
          86
          96
       7 7
          17
          27
          37
          47
          57
          67
          77
          87
          97
       8 8
          18
          28
          38
          48
          58
          68
          78
          88
          98
       9 9
          19
          29
          39
          49
          59
          69
          79
          89
          99
-------------------------------------------------------------

 A gentleman informed me of p-adics in AMERICAN MATHEMATICAL MONTHLY
vol 98, 1991, April pp 355-368. I wrote back that I had glanced it
some years back.

 There was nothing in it that would help me with Geometry/Space =
Numbers. Ask me my opinion on Fractals and I would say that it is a
present day fad or fashion. Ask me what I think fractals are worth and
I would say they exist but of small importance and that those that
spend a-lot of time with Fractals are in a state of "Misplaced
Importance".

  So, I gave this article another look yesterday. I am surprized that a
High School math teacher can discuss p-adics for more than 2 minutes. I
had it in mind that if all High School math teachers in the USA and
Canada were asked to talk about p-adics, only a few would have ever
heard of them and none of them could talk about p-adics for more than 2
minutes. Perhaps Mr. Albert Cuoco of Woburn MA is some special type of
High School teacher. I do not know.

  Anyway reviewing this article. Quoting.

p355 " Pictures that inspire images of p-adic objects are rare (see the
frontispiece of [3] for an example). In this note, we show how to
represent, for any prime p, the elements of Z_p as the points on a
fractal R2. The resulting correspondence captures many of the important
algebraic and topological facets of Z_p."

 "...Sierpinski triangle..."

              /\
              --
           /\ /\
           -- --

   /\ /\
   -- --
/\ /\ /\ /\
-- -- -- --

Now the issue I need to raise is that I can see the sawed-off bottoms
and tops of the tangent sinusoids of the first quadrant by these
triangles.

Or, even the trees above of p-adics I can visualize the sawed-off
bottoms and tops of the tangent sinusoids.

  I am not at all surprized that p-adic pictures, as Mr. Cuoco says are
rare. I am glad they are rare. Because if they had been better
researched before I came along in 1993, someone else would have
realized that Lobachevskian Geometry is P-adic Numbers and that
Riemannian Geometry is the Doubly Infinite Numbers.

  So, you can visualize that Sierpinski triangle as rotated, rotate it
so that it is the first quadrant. And instead of the p-adics as smaller
triangles visualize them as the cut-off branches of the tangent
sinusoid.

  And those trees above such as this one:

> 0
> / \
> 0 1
> / \ / \
> 0 2 1 3
> / \ / \ / \ / \
> 0 4 2 6 1 5 3 7

  are points of those sawed-off branches of the tangent sinusoid.

  Remember though, that the N-adic Numbers are richer, far richer than
the P-adics.

   Is P-adic Numbers = Lobachevskian Geometry ?

or

   Is P-adic Numbers = Riemannian Geometry ?

 The tangent sinusoid model is no model at all and I abandoned it
in the summer of 1997.

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